Chapter 4: Problem 36
In Exercises \(35-38\) , find the function with the given derivative whose graph passes through the point \(P\) . $$f^{\prime}(x)=\frac{1}{4 x^{3 / 4}}, \quad P(1,-2)$$
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$$ \begin{array}{l}{\text { Multiple Choice Which of the following functions is an }} \\ {\text { antiderivative of } \frac{1}{\sqrt{x}} ? \quad \mathrm{}}\end{array} $$ $$ (\mathbf{A})-\frac{1}{\sqrt{2 x^{3}}}(\mathbf{B})-\frac{2}{\sqrt{x}} \quad(\mathbf{C}) \frac{\sqrt{x}}{2}(\mathbf{D}) \sqrt{x}+5(\mathbf{E}) 2 \sqrt{x}-10 $$
Newton's Method Suppose your first guess in using Newton's method is lucky in the sense that \(x_{1}\) is a root of \(f(x)=0 .\) What happens to \(x_{2}\) and later approximations?
Growing Sand Pile Sand falls from a conveyor belt at the rate of 10 \(\mathrm{m}^{3} / \mathrm{min}\) onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 \(\mathrm{m}\) high? Give your answer in \(\mathrm{cm} / \mathrm{min.}\)
Writing to Learn You have been asked to determine whether the function \(f(x)=3+4 \cos x+\cos 2 x\) is ever negative. (a) Explain why you need to consider values of \(x\) only in the interval \([0,2 \pi] . \quad\) (b) Is f ever negative? Explain.
Oscillation Show that if \(h>0,\) applying Newton's method to $$f(x)=\left\\{\begin{array}{ll}{\sqrt{x},} & {x \geq 0} \\ {\sqrt{-x},} & {x<0}\end{array}\right.$$ leads to \(x_{2}=-h\) if \(x_{1}=h,\) and to \(x_{2}=h\) if \(x_{1}=-h\) Draw a picture that shows what is going on.
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